Dynamic behaviour

response of RC panels, thanks to the tangent approach which allows to take into account plastic strains in case of cyclic loads.

n

ωξn

β = 2^⋅

(4.46) The mass-proportional damping, expressed in Eq.(4.43), and the stiffness-proportional damping, expressed in Eq.(4.46) are plotted in Figure 4.21-a. As shown in Figure 4.21-a the mass-proportional damping allows to damp the low-frequency modes while the stiffness-proportional damping allows to damp the high-frequency modes and it is quite important in NLFEA because it can provide sufficient dissipation to suppress the high-frequency numerical noise. Nevertheless, by themselves neither of the two damping models are appropriate for practical application of Multi Degrees of Freedom (MDF) systems.

The Rayleigh damping is introduced as the algebraic sum of the mass-proportional and the stiffness-proportional damping on the base of Eq.(4.47):

n n

n M K

C =α⋅ +β⋅ (4.47)

By substituting Eq.(4.42) into Eq.(4.47) the Rayleigh damping coefficient can be derived as a function of α ^and β as expressed in Eq.(4.48) and shown in Figure 4.21-b.

n n

n β ω

ω ξ = α ⋅ + ⋅

2 1

2 (4.48)

In that case the coefficients α ^and β can be determined by specifying damping ratio ξ for the ith and jth modes, as illustrated in Figure 4.21-b. The solution of the obtained system leads to the following equation:

j i

j i

ω ω

ω ξ ω

α +

⋅ ⋅

⋅

=2 (4.49)

j

i ω

ξ ω β = ⋅ +2

(4.50)

(a) (b)

Figure 4.21 – Variation of damping ratios with natural frequency: (a) mass-proportional and stiffness-proportional damping and (b) Rayleigh damping.

ωn

ξn

2

n n

β ω ξ = ⋅

n

n αω

ξ = ⋅

Mass-proportional 2

ωn

ξn

2 2

n n n

β ω ω

ξ α + ⋅

= ⋅ ξ

ωi ωj

4.2.2.2 Implementation of Rayleigh damping in PARC_CL 2.0

ABAQUS code [Abaqus 6.12, 2012] allows the definition of the mass-proportional damping, α ^{, as an} input value for the nonlinear finite element analysis. On the other hand, due to the fact that the stiffness matrix is modified by means of a user defined subroutine (the PARC_CL 2.0 crack model) it is not possible to define the stiffness-proportional damping, β , as an input value. For this reason, in order to consider the energy dissipation due to the material behaviour, it was necessary to introduce the stiffness proportional damping in the PARC_CL 2.0 crack model.

The stiffness-proportional Rayleigh damping is introduced in the PARC_CL 2.0 according to the following Eq.(4.52):

ε β

σ_damp= ⋅E ⋅′ & _(4.51)

The damping contribute on the concrete stress vector can be calculated as follow:

{ }















′ ⋅

⋅

′ ⋅

⋅

′ ⋅

⋅

=

12 12

2 2

1 1 2

, 1

γ β

ε β

ε β σ

&

&

&

G E E

c c

damp (4.52)

In the same way the damping contribute on the steel stress vector can be calculated as follow:

{ }













 ⋅ ′ ⋅

=

0

, 0

xi si

y damp x

E

i i

ε β σ

&

(4.53)

The overall dynamic concrete stress vector, {σ^1,2}dyn , is calculated as the sum of the static contribution of the material, {σ^1,2}, already calculated in Eq.(4.30) and the damping contribution, {σ^{1,2}damp ,} just calculated according to Eq.(4.52):

{ } { } { }

σ1,2 _dyn= σ1,2 + σ1,2 _{damp (4.54)}

Similarly, the overall dynamic steel stress vector, {σ^xi,yi}dyn , is calculated as the sum of the static contribution of the material, {σ^xi,yi}, as calculated in Eq(4.31) and the damping contribution, {σ^xi,yi}damp ,

reported in Eq.(4.53):

{ } { } { }

σ^xi,^yi _dyn = σ^xi,^yi + σ^xi,^yi _{damp (4.55)}

Both the concrete and each steel dynamic stress vector can be transformed from their local coordinate system to the overall global x,y coordinate system using respectively Eq.(4.56) and Eq.(4.57):

{ } [ ] { }

dyn t

dyn y c

x T _1,2

, , σ

σ = _ψ ⋅ (4.56)

{ } [ ] { }

x y _dyn t

dyn i i y s

x T _i,i

,

, , σ

σ = _θ ⋅ (4.57)

The total dynamic stress in the x,y-system is obtained by assuming that concrete and reinforcing bars behave like two springs placed in parallel:

{ } { } ∑ { }

=

+

= ⁿ

i

dyn i y s x dyn i

y c dyn x y x

1

, , , , ,

, σ ρ σ

σ (4.58)

where n is the total number of the orders of bars.

In the finite element implicit calculation, using Newton-Raphson’s algorithm to solve the balance equations, the tangent stiffness modulus is needed. The tangent stiffness modulus, for the damping contribution can be expressed, on the base of Eq.(4.51), as follow:

E dt d

d _dampε =β⋅ ⋅′ 1

σ (4.59)

where dt denotes the increment of time.

The Jacobian tangent stiffness matrix for static contribution, already reported in Eq.(4.36) for concrete and Eq.(4.37) for steel, contains itself the tangent stiffness modulus. As a consequence of this the Jacobian tangent stiffness matrix related to the damping contribution can be express as in Eq.(4.60) for concrete and in Eq.(4.61) for steel:

[ ]

⁼

[ ]

^⋅_^{ ⋅ }_^

D dt

D _damp 1

2 , 1 2

,

1 β (4.60)

[ ]

⁼

[ ]

^⋅_^{ ⋅ }_^

D dt D_{i i xiyi}

y damp x

1

,

, β (4.61)

where [D1,2] and [Dxi,yi] represent the static Jacobian matrix for concrete and steel as reported in Eq.(4.36) and Eq.(4.37).

Finally, the overall dynamic Jacobian matrix can be calculated as the sum of static and damping contribution:

[ ] [ ] [ ] [ ]

^



 



 + ⋅

⋅

= +

= D D D dt

D _{dyn damp} 1

2 1

, 1 2

, 1 2 , 1 2

,

1 β (4.62)

[ ] [ ] [ ]

^{= + =}

[ ]

^⋅_^{ + ⋅ }_^

D dt D

D

Dxi yi dyn xi yi xiyi damp xiyi

1 1

, ,

,

, β ^(4.63)

4.2.2.3 Validation of the model, simple 1 DOF model

In order to validate the proposed model and to assess its capability to predict the damping effect on a RC member, a comparison between NLFEA and analytical formulation is presented.

A Single Degree of Freedom (SDF) system, represented by the cantilever tower of Figure 4.22-a, was used as a reference system. The cantilever tower is characterized by a section of 100x100 mm and a height of 1000 mm. The system is fixed at the base and it presented a lumped mass at the top equal to 50 Kg. The elastic modulus of the material was set equal to 28000 MPa.

Figure 4.22 – Free vibration oscillator: (a) SDF analytical system and (b) NLFE model.

4.2.2.3.1 Analytical formulation

The cantilever tower of Figure 4.22-a was subjected to an horizontal displacement at the top equal to 0.01 mm, such as to avoid cracking and to keep the material in the elastic field. After the top mass was released, the system was subjected to free vibration based on the dynamic equation of motion reported in Eq.(4.64):

=0

⋅ +

⋅ +

⋅u C u m u

K & && ^(4.64)

where K represents the stiffness of the SDF system, C the damping coefficient and M the mass applied to the system. u is the horizontal displacement as a function of time and an over dot represents a time derivative.

Being M the mass applied on the system, equal to 50Kg, the stiffness, K, and the damping coefficient, C, can be derived as follow:

M

K =ω²⋅ (4.65)

M C=2⋅ξ⋅ω⋅

where ω represents the natural frequency of the system, expressed by Eq.(4.66), and ξ represents the damping ratio.

⋅ f

⋅

= π

ω 2 (4.66)

where f is the frequency of the system.

To solve the analytical dynamic equation of motion, the frequency derived from NLFEA, equal to 19.7 Hz, is used. This assumption permits to compare analytical and NLFEA results, with the same basic hypotheses. It is still important to underline that the frequency deduced from NLFEA is quite similar to the analytically calculated one (fanalitycal=18.8 Hz).

Free vibration for un-damped oscillator

The analytical solution of Eq.(4.64) for un-damped system can be derived by imposing C equal to 0 and leads to the following equation:

t C

t C

t

u( )= ₁⋅cosω + ₂⋅sinω (4.67)

where C1 and C2 are the constants of integration and can be obtained by imposing the boundary conditions to the system:

mm u

u_(t=0)= ₀ =0.01 → C₁=u₀ =0.01mm (4.68)

0 0

) 0

(₌ =u =

u&_t & → ₂ = ⁰ =0 ω C u&

(4.69) Finally the equation of motion, for the analysed un-damped case study, can be expressed as in Eq.(4.70). Eq.(4.70) is graphically plotted in Figure 4.23-a.

t u

t

u( )= ₀⋅cosω (4.70)

Free vibration for damped oscillator

By solving the dynamic equation of motion reported in Eq.(4.64), the analytical solution of a damped system subjected to free vibration can be derived as follow:

(

^{C t C t}

)

e t

u()= ^−νωt⋅ ₁⋅cosω_d + ₂⋅sinω_d (4.71)

where ω^d represents the damped frequency and it can be calculated according to Eq.(4.72). C1 and C2 are the constants of integration and it can be derived by imposing the boundary conditions to the system as reported in Eq.(4.73) and Eq.(4.74).

1 ξ2

ω

ω_d = ⋅ − (4.72)

mm u

u_(t=0)= ₀ =0.01 → C₁=u₀ =0.01mm (4.73)

0 0

) 0

(₌ =u =

u&_t & →

d

u C u

ωω

υ 0

0 2

⋅

⋅

= & +

(4.74)

Finally the equation of motion, for the analysed damped case study, can be expressed as in Eq.(4.75).

Eq.(4.75) is graphically plotted in Figure 4.23-b.







 ⋅ + ⋅ ⋅ ⋅

⋅

= ⁻ u t

t u

e t

u _d

d d

t ω

ω ω ω υ

νω cos sin

)

( ₀ ⁰ (4.75)

In can be observed that for an un-damped system (ξ=0), the Eq.(4.75) is reduced to Eq.(4.70).

(a)

(b) Figure 4.23 – Free vibration oscillator analytical response: (a) un-damped system (b) damped system.

4.2.2.3.1 Nonlinear finite element analysis

NLFEA was carried out in order to assess the proposed damping model in the PARC_CL2.0 crack model. The same cantilever tower, representing a SDF system, reported in Figure 4.22-a was modelled by means of NLFEA as shown in Figure 4.22-b.

The height of the tower was subdivided into 10 shell elements with 4 nodes and 4 Gauss integration points (defined S4 in ABAQUS code [Abaqus 6.12, 2012]), so each element presents a 100x100 mm size in plane. The thickness of each element was set equal to 100 mm in order to achieve the same cross section as the reference analytical specimen. The two nodes at the base of the tower were fixed and at the top of the tower two mass elements of 25 Kg were added.

In order to highlight the frequency of the numerical system, a frequency analysis was carried out before running the dynamic analysis. The frequency analysis highlight that the first vibration mode, able to excite the

-0.01 0 0.01

0 0.25 0.5 0.75 1 1.25 1.5 t [sec]

u(t) [mm]

Analytical formulation - Eq.(3.69)

-0.01 0 0.01

0 0.25 0.5 0.75 1 1.25 1.5

ξ=1%

ξ=2%

ξ=5%

t [sec]

u(t) [mm]

Analytical formulation - Eq.(3.74)

99.4% of the mass, presents a frequency, f=19.7 Hz, so that the corresponding natural frequency was equal to ω=123.8 Hz.

This frequency was used to calculate the stiffness-proportional damping coefficient, introduced in the PARC_CL 2.0 crack model, according to the formulation proposed in Eq.(4.46).

Due to the fact that the analysed system presents a single degree of freedom, it was necessary only a single frequency to calibrate the damping coefficient. In order to validate the stiffness-proportional damping introduced in the PARC_CL 2.0 crack model, only the stiffness-proportional damping coefficient was calculated.

The NLFE model was firstly subjected to an horizontal displacement at the top of the specimen equal to 0.01 mm. In a second phase the system was released to experience free vibration in dynamic field.

The time step was set equal to 0.0005 sec and the numerical solution is obtained by means of the Newmark’s implicit method considering average acceleration.

In Figure 4.24 are presented the comparison between NLFEA and analytical formulation for different values of damping ratio.

The results reported in Figure 4.24 show that the stiffness-proportional damping introduced in the PARC_CL 2.0 crack model allows to reproduce well the dissipation of energy due to damping in a linear elastic system.

-0.01 (a)

0 0.01

0 0.25 0.5 0.75 1 1.25 1.5

Analytical formulation - Eq.(3.69) NLFEA

t [sec]

u(t) [mm] ξ=0

-0.01 0 0.01

0 0.25 0.5 0.75 1 1.25 1.5

Analytical formulation - Eq.(3.74) NLFEA

t [sec]

u(t) [mm] ξ=1%

(c)

(d) Figure 4.24 – Free vibration, comparison between NLFEA and analytical formulation for different values of

damping ratio: (a) un-damped system, (b) ξ^{=1%, (c)} ξ=2% and (d) ξ^=5%.

Nel documento Implementation of PARC_CL 2.0 crack model for reinforced concrete members subjected to cyclic and dynamic loading (pagine 121-129)